Is there a way to make this curve fit my data better? I'm working off Safe Leads and Lead Changes in Competitive Team
Sports and trying to apply the same
techniques to baseball scores, using number of outs remaining instead
of the number of seconds. I'm pretty far out of my depth reading the
paper so I'm just cherry-picking the algorithms that look right and
trying to use them with baseball data. I get a pretty reasonable curve
(similar to Figure 16 in the paper), but as the game gets more safe,
the erf(v) curve overestimates the probability the lead is safe by a
few percentage points (I wouldn't mind if it underestimated a
little). Here's a picture: the red triangles are the observed values,
and the black line is the erf(z) curve; the hand-drawn blue line is roughly what I wish the erf(z) curve looked like.

Is there a way to make my curve fit the data better, either by applying some
additional transformation to the result of erf(z) or using something
other than erf in the first place? Preferably also something easy to
implement in Java (for my Android app).

If you want to examine the data yourself:


*

*Data used to make the graph (data from 2000-2014 run through the program with -p 50)

*Program used to create the graph (with some instructions
in its comments).

 A: It's not a good idea to take data and say "what functions might fit this" -- you can't account for the "researcher degrees of freedom" in this.
Worse, when you ask large numbers of experts you're potentially adding a large number of potential "guesses" but the only ones you will hear about are the "ones that work", so you can't hope to guess at the effective degrees of freedom used by asking our thoughts on your data.
(A less biased way to do this would be to randomly choose a fraction of your points, post those, then fit the parameters and assess the fit on other parts of the data.)
That said, by eye one can see that $1-\exp(-\theta\:\! x)$ will be a better fit than what you have (looking more closely at the picture, it looks like $\theta\approx 1.8$, or possibly a bit lower, would fit adequately). 
However, until you can confirm the fit on data not used to identify or fit the curve, you should take the quality of the fit with a grain of salt; it will fit well but could have relatively poor out of sample performance. 
(It's also not clear to me this is a suitable way to model data of this sort but I haven't looked at the paper nor at your data set.)
